Literal Equation Calculator
This literal equation calculator rearranges a formula to make any letter the subject, and names the operation it undoes at every step.
Type A = 1/2*h*(b1 + b2), h and you get h = 2A/(b₁ + b₂), with each undo written out.
Solving literal equations means exactly this: isolate one letter and express it in terms of the others. The word is older than it looks — "literal" comes from littera, Latin for letter, so a literal equation is simply one built out of letters rather than numbers.
It Gives Back a Formula, Not a Number
That is the whole difference. Solving 3x − 7 = 8 gets you x = 5. Rearranging A = πr² for r gets you r = √(A/π), and there is no number anywhere in it. That is what makes literal equations feel unfamiliar even to students who solve numeric ones without thinking.
Which is why people find these harder than they should. The skill is identical to solving an equation — do the same thing to both sides until the letter is alone — but with no arithmetic to check yourself against, there is nothing to tell you when a step has gone wrong.
Work From the Outside In
Look at your letter and ask what is wrapped around it. Whatever was applied last is what you undo first. Teachers call the goal making that letter the subject of the formula; older textbooks call the whole business transposing formulae. Both mean isolate it.
Take v = u + at, solving for t. The t has an a multiplying it, and a u added on outside that. So the u comes off first: v − u = at. Then divide by a: t = (v − u)/a. Two steps, in that order.
Try it the other way round — dividing by a first — and you get v/a = u/a + t, which is not wrong but is now further from the answer than when you started. That is the mistake that makes this topic feel impossible.
Two Awkward Cases Worth Recognising
The first is when your letter is the thing being subtracted. In a − x = b, adding and subtracting your way to x tangles the signs. Swap the sides instead: x = a − b, done in one move.
The second is when the letter is underneath a fraction. In a/x = b nothing can be peeled off the top, because x is not on the top. Turn both sides over and it becomes x/a = 1/b, so x = a/b.
Both come up constantly and both look worse than they are.
| Formula | Make this the subject | Result |
|---|---|---|
| A = ½h(b₁ + b₂) | h | h = 2A/(b₁ + b₂) |
| F = 9/5 C + 32 | C | C = 5(F − 32)/9 |
| A = πr² | r | r = √(A/π) |
| 1/f = 1/u + 1/v | u | u = 1/(1/f − 1/v) |
The Plus-or-Minus Nobody Can Decide For You
A = πr² rearranges to r = √(A/π). Strictly, r = ±√(A/π), because both signs square to the same thing.
But r is a radius. A radius of −3 is not a smaller circle, it is nothing at all. So the negative root is discarded — not by algebra, but because of what the letter stands for.
That judgement cannot be made from the symbols. A page that silently drops the ± is right about half the time and teaching you nothing either way, so this one shows the ± and says whose decision it is.
The rule of thumb is short. If the letter measures how much of something there is — a length, a radius, a mass, a time, a speed — it cannot be negative and the positive root is your whole answer. If it measures a position or a direction — a coordinate, a velocity, a temperature — it can be negative, and dropping the minus root throws away a real solution.
Examiners know this, which is why "give both values of r" and "give the radius" are marked differently on the same working. The algebra is identical. Only the sentence around it changes.
Leave the Letter Off and Get Them All
Type P = 2*(l + w) with no comma and you get every rearrangement at once: l = P/2 − w, and w = P/2 − l.
Useful more often than it sounds. Half the time you do not yet know which form you need — you are checking whether a formula can be turned round at all, or you want to see the shape of all of them before choosing.
When the Letter Appears Twice
A = P + Prt is the one every textbook sets, and most calculators refuse it. The P is in two places, so there is no outer operation to peel.
Collecting fixes it. Both terms carry a P, so factor it out: A = P(1 + rt), and one division finishes the job — P = A/(1 + rt). Same with a letter on both sides of the equals: ax + b = cx + d gives x = (d − b)/(a − c). This page does that collection for you and shows the factored line.
It stops where collecting genuinely stops. In S = 2πr² + 2πrh the r is squared in one term and not the other, so there is nothing common to take out — that one is a quadratic in r and belongs to the quadratic formula. A letter underneath a division, or inside a root in one place and outside it in another, stops for the same reason. Solving that surface-area formula for h is fine, because h appears once.
Most competitors draw the line much earlier. One of the better-known ones states plainly that it handles literal equations "with no powers" at all, which rules out the area of a circle.
How to Use This Literal Equation Calculator
The letters have to come from your own keyboard, so tap My keyboard above the pad and type them there — the pad handles the brackets, powers and operations. Write the formula, then a comma, then the letter: A = 1/2*h*(b1 + b2), h. Use * between things being multiplied, ^ for a power, and sqrt( for a root. Subscripts work as b1 and b2.
Every step names the operation it undid. If your equation has a single unknown and everything else is a number, that is a linear equation instead. If the letter you want sits in an exponent, getting it out needs logarithms and belongs on the exponential equation page. And when rearranging leaves you with a squared term, the quadratic formula finishes it.
Literal Equation Calculator FAQ
What is a literal equation?
What does "make x the subject" mean?
Which operation do I undo first?
What if my letter is being subtracted?
What if the letter is on the bottom of a fraction?
Do I keep the ± when a square root comes off?
Can a letter that appears twice still be made the subject?
Is this the same as solving an equation?
Outside in, both sides every time, and stop when the letter is alone on one side and absent from the other.
